CHARMM-G, a GPU based MD Simulation code with PME and Reaction Force field for

Studying Large Membrane Regions

Narayan Ganesan1, Sandeep Patel2, and Michela Taufer1

Computer and Info. Sciences Dept.1

Chemistry and Biochemistry Dept.2

University of Delaware

Outline

• Overview of forces in molecular dynamics

• Data structures and methodology

• PME for long distance electrostatic interactions

• Steps involved in PME calculations

• Performance and profiling of large membranes

• Related work and conclusions

Classical Forces

Classical Forces

Bonded

BondsAngles

Dihedrals

Non-Bonded

ElectrostaticVan der Waals

Reaction Field (RF) PME

Bond Interactions

• Bond forces: acts only within pairs of molecules

• Angle forces: acts only within a triad of atoms

• Torsion or dihedral forces: acts only within quartet of atoms

Non-bond Interactions:Van der Waals Potential

• Van der Waals or Lennard-Jones potential: Decays rapidly with distance

612

4rr

EVDW

• A cutoff of ~10A, accurately captures the effect of the Van der Waals potential

Non-Bond Interactions:Electrostatic Potential

• Coulomb Potential: inverse square law

€

Ucoulomb =1

4πε 0

q1q2

| r1 − r2 |

221

21

0 ||4

1

rr

qqFcoulomb

• Decays as 1/r with distance• Since 1/r decays rather slowly,

the potential can act over long distances

• Choosing a cutoff for electrostatic force/potential causes computational errors and inaccuracy • Our solutions to sum long distance electrostatic forces:

Reaction Force Field (RF) Ewald summation / Particle Mesh Ewald (PME)

GPU Implementation: Data Structures

• A single thread is assigned to each atom• For each atom a set of lists is maintained:

Bond list stores list of bonds the atom belongs to Angle list stores list of angles the atom belongs to Dihedral list stores list of dihedrals the atom belongs to Nonbond list stores non-bond interactions with atoms within cutoff

q2, r2q6, r6q8, r8q9, r9

q1 q2 q3

q9q7 q8

q5 q6q4

Nonbond list for q5:

MD Simulation

• MD simulations are iterative executions of MD steps• Each iteration computes forces on each particle due to:

Bonds – Bond List Angles – Angle List Dihedrals – Dihedral List Electrostatic Van der Waals

• If Ewald summation is used an additional component is added: Long distance interaction using PME method

- Nonbond List

Bond, angle, and dihedral lists are unchanged for each atom throughout the simulation

Nonbond list is updated based on a cutoff buffer

Ways to Update Nonbond List

• Global neighbor list Each thread can iterate through the

global list of atoms to build the nonbond list

• Cell-based neighbor list Divide the domain into equal cells of

size = cutoff Search only in current cell and

adjacent cells for neighboring atoms There are 26 adjacent cells and 1

current cell in 3-dimensions

• Cell-based list is computationally very efficient but also needs regular cell updates

Cell Updates

• Single thread manages a single or a set of cells• Each cell is managed by a list of

atoms in the cell called ‘CellList’• When an atom ‘i’ moves from

Cell A to Cell B, the thread responsible for Cell A updates the list of Cell B via thread safe integer atomic intrinsics• Invalid atoms are removed

from the cell lists by the ‘CellClean’ kernel

q1 q2 q3

q9q7 q8

q5 q6q4

Periodic Boundary Condition

q1 q2 q3

q9q7 q8

q5 q6q4

q1 q2 q3

q9q7 q8

q5 q6q4

q1 q2 q3

q9q7 q8

q5 q6q4

q1 q2 q3

q9q7 q8

q5 q6q4

q1 q2 q3

q9q7 q8

q5 q6q4

q1 q2 q3

q9q7 q8

q5 q6q4

q1 q2 q3

q9q7 q8

q5 q6q4

q1 q2 q3

q9q7 q8

q5 q6q4

Cell of interest of edge vectors ax, ay

Region of influence

Reaction Force Field

• Any molecule is surrounded by spherical cavity of finite radius Within the radius, electrostatic interactions are calculated explicitly Outside the cavity, the system is treaded as a dielectric continuum

• This model allows the replacement of the infinite Coulomb sum by a finite sum plus the reaction filed

where the second terms is the reaction filed correction and Rc is the radius of the cavity

€

Uc =1

4πε 0

qiq ji< j∑ 1

rij−B0rji

2

2Rc3

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Coulomb potential

Ewald Summation Method (I)

• Proposed by Paul Peter Ewald in 1921 for crystallographic systems• Has found applications in molecular, astrophysical and

crystallographic systems• Used to sum inverse distance potential over long distance

efficiently – e.g., Gravity and Coulomb Potential.• Was started to be used in the late 70s for numerical

simulations O(NlogN) instead of O(NxN)

Ewald Summation Method

• Three contributions to the total energy, depending on the distance of the interaction: Direct space (Edir)

Reciprocal space (Erec)

Self energy (Eself )

Ewald Summation Method (II)

• Divide interactions into short range (Direct Space) and long range (Reciprocal Space)

Short Range

Long Range

Direct space using Nonbond List

Fourier Space

||4

|)|(erfc

10 ji

jijidir rr

rrqqE

)()()/(exp

2

1

02

222

mSmSm

m

VE

mrec

V - Volume of the simulation region

S(m) – Structure parameters

Steps in SMPE

Put charges on grids1

FFT of charge grid

2

Multiply with structure constants

3

FFT back

4

€

∂U∂ri

Compute force on atom i by calculating

Convolution yields potential at grid points which have to be summed

5

Charge Spreading

• Each charge is spread on a 4x4x4 = 64 grid points in 3-D Grid spacing 1 A by a cardinal B-Spline of order 4 Create a 3 dimensional Charge Matrix “Q”.

• Mesh-based charge density Approximation by sum of charges at each grid point Multiple charges can influence a single lattice point

Charges

ni

iiii kzMkyMkxMqkkkQ..1

342414321 )()()(),,(

Essman et al., J. Chem. Phys. 1995

xi yi zi: position of the ith charge; k1 k2 k3: index of the lattice point

Cardinal B-Spline of Order 4

• B-Spline has a region of influence of 4 units Each unit = 1A

• During charge spreading B-Spline has an impact on the neighboring 4x4x4 cells in 3 dimensions

CPU vs. GPU Charge Spreading

• Charge Spreading by a cardinal B-Spline of order 4:

• CPU implementation is straightforward Time computation: Natoms x 4 x 4 x 4 time steps

• GPU implementation is hard to parallelize Can lead to racing conditions - need floating point atomic writes

• Current version of CUDA supports atomic writes for integers only Charges need to be converted to fixed point in order to utilize the

functionality

Unit cellcharges

CPU vs. GPU Charge Spreading

CPU spreading of charges: GPU gathering of charges by a cardinal B-Spline of order 4:

• Charge spreading on GPU can be parallelized easily by the grid points instead of the atoms

• Each thread works on a single or a set of grid points

• Need O(ax*ay*az) threads, with each thread parsing through all the atoms within 4x4x4 neighborhood –> O(N)

Each thread is assigned to a lattice point

GPU Charge Spreading (I)

• Each lattice point maintains a list of atoms within 4x4x4 neighborhood for charge gathering

1

3

2

Neighbor list of point: q1, r1q2, r2q3, r3

Effect of charges 1, 2, 3 are gathered at the lattice point

GPU Charge Spreading (II)

2

2’

• When a charge moves, several lattice points need to be updated• The charge is added to the neighbor

list of lattice points in dark gray• The charge is removed from the

neighbor list of lattice points in light gray• Lattice points in white are not affected• Since there are equal number of light

gray and dark gray lattice points, a 1-to-1 mapping was devised

The threads for lattice points in light gray update the list of lattice points in dark gray in a 1-to-1 fashion

GPU Charge Spreading (III)

2

2’

1

1’

• When a single lattice point is updated by multiple threads, thread safe integer atomic intrinsics are used to update the cell lists

Fast Fourier Transform

• CUFFT provides library functions to compute FFT and inverse FFT 3D FFT implemented with series of 1D FFTs and

transpositions• CUFFTExec can be optimized by choosing proper FFT

dimensions Power of 2

Scientific Challenge

• One-third of the human genome is composed of membrane-bound proteins • Pharmaceuticals target membrane-bound protein receptors

e.g., G-protein coupled receptors Importance of systems to human health and understanding of

dysfunction

• State-of-the-art simulations only consider small regions (or patches) of physiological membranes • Heterogeneity of the membrane spans length scales much

larger than included in these smaller model systems. • Our goal: apply large-scale GPU-enabled computations for the

study of large membrane regions

DMPC

• DiMyristoylPhosphatidylCholine (DMPC) lipid bilayers

Explicit solvent i.e., water

Membrane152A

92A

92A

Small system: 17 004 atoms, 46. 8A x 46.8 A x 76.0 A

Large system: 68 484 atoms, 93.6 A x 93.6 A x 152.0 A

Performance

Small membrane (17 004 atoms)

Case studies: Global neighbor list and RF (I), with cell-based list and RF (II), with neighbor list and PME (III), and with cell-based neighbor list and PME (IV)

Large membrane (68 484 atoms)

Kernel Profiling (I)

• Large membrane – RF methodGlobal neighbor list Cell-based neighbor list

Kernel Profiling (II)

• Large membrane – PME methodGlobal neighbor list Cell-based neighbor list

Related Work

• Other MD code including PME method:M. J. Harvey and G. De. Fabritiis, J. Chem. Theory and Comp, 2009”

• Our implementation is different in terms of: Charge spreading algorithm Force field methods, including RF

Conclusions and Future Work

• CHARMM-G is a flexible MD code based on the CHARMM force field integrating Ewald summation Reaction force field

• The code supports explicit solvent representations and enables fast simulations of large membrane regions• Improvements of the CUDA FFT will further improve the

performance presented in the paper• Future work include:

Code optimizations and parallelization across multiple GPUs Scientific characterization of large membranes

32

Acknowledgements

Sponsors:

Collaborators:Sandeep Patel, Brad A. Bauer, Joseph E. Davis (Dept. of Chemistry, UD)

Related work:Bauer et al, JCC 2010 (In Press)

Davis et al., BICoB 2009

More questions: taufer@udel.edu

GCL Members:

Trilce Estrada Boyu Zhang

Abel Licon Narayan Ganesan

Lifan Xu Philip Saponaro

Maria Ruiz Michela Taufer

GCL members in Spring 2010